Extension field
F is a field.
A field E for which F  E and for which the operations
of F are those of E restricted to F.
Fundamental Theorem of Field Theory
(Kronecker’s Theorem)
Let F be a field and f ( x) a nonconstant polynomial in
F [ x] . Then there is an extension field E of F in which
f ( x) has a zero.
f ( x) splits in E
E is an extension field of F.
f ( x) can be factored as a product of linear factors in
E[ x] .
Splitting field for f ( x) over F
F is a field.
An extension field E of F in which f ( x) splits, but for
which f ( x) does not split in any proper subfield of E.
Existence of Splitting Fields
Let F be a field and let f ( x) be a nonconstant element
of F [ x] . Then there exists a splitting field E for f ( x)
over F.
F (a)  F[ x]/  p( x) 
Let F be a field and p( x)  F[ x] be irreducible over F.
If a is a zero of p( x) in some extension E of F, then
F (a) is isomorphic to F[ x]/  p( x)  . Furthermore, if
deg p( x)  n , then every member of F (a) can be
uniquely expressed in the form
cn1an1  cn2an2  ...  c1a  c0
where c0 , c1,..., cn1  F .
F (a)  F (b)
Let F be a field and p( x)  F[ x] be irreducible over F.
If a is a zero of p(x) in some extension E of F and b is a
zero of p(x) in some extension E’ of F, then the fields
F (a) and F (b) are isomorphic.
Lemma, p. 351
Let F be a field, let p( x)  F[ x] be irreducible over F,
and let a be a zero of p( x) in some extension of F. If
 is a field isomorphism from F to F’ and b is a zero of
 ( p( x)) in some extension of F’, then there is an
isomorphism from F (a) to F '(b) that agrees with  on
F and carries a to b.
Extending  : F  F '
Let  be an isomorphism from a field F to a field F’ and
let f ( x)  F[ x] . If E is a splitting field for f ( x) over F
and E’ is a splitting field for  ( f ( x)) over F’, then there
is an isomorphism from E to E’ that agrees with  on F.
Splitting Fields Are Unique
Let F be a field and let f ( x)  F[ x] . Then any two
splitting fields of f ( x) over F are isomorphic.
Derivative
Let f ( x)  an xn  an1xn1  ...  a1x  a0 belong to F [ x] .
The polynomial f '( x)  nan xn1  (n 1)an1xn2  ...  a1
in F [ x] .
Properties of the Derivative
Let f ( x), g ( x)  F[ x] and let a  F . Then
1. ( f ( x)  g ( x))'  f '( x)  g '( x) .
2. (af ( x))'  af '( x) .
3. ( f ( x) g( x))'  f ( x) g '( x)  f '( x) g( x)
Criterion for Multiple Zeros
A polynomial f ( x) over a field F has a multiple zero in
some extension E if and only if f ( x) and f '( x) have a
common factor of positive degree in F [ x] .
Zeros of an Irreducible
Let f ( x) be an irreducible polynomial over a field F. If
F has characteristic 0, then f ( x) has no multiple zeros.
If F has characteristic p  0 , then f ( x) has a multiple
zero only if it is of the form f ( x)  g ( x p ) for some
g ( x)  F[ x] .
Perfect field
A field F with characteristic 0 or with characteristic p
and F p  {a p | a  F}  F .
Finite Fields Are Perfect
Every finite field is perfect.
Criterion for No Multiple Zeros
If f ( x) is an irreducible polynomial over a perfect field
F, then f ( x) has no multiple zeros.
Zeros of an Irreducible over a Splitting Field
Let f ( x) be an irreducible polynomial over a field F and
let E be a splitting field of f ( x) over F. Then all the
zeros of f ( x) in E have the same multiplicity.
Factorization of an Irreducible over a Splitting Field
Let f ( x) be an irreducible polynomial over a field F and
let E be a splitting field of f ( x) . Then f ( x) has the
form a( x  a1)n ( x  a2 )n...( x  at )n , where a1, a2 ,..., at are
distinct elements of E and a  F .